On the Fourth Power Mean of the Two-Term Exponential Sums

The main purpose of this paper is to use the analytic methods and the properties of Gauss sums to study the computational problem of one kind fourth power mean of two-term exponential sums and give an interesting identity and asymptotic formula for it.


Introduction
Let ≥ 3 be a positive integer. For any integers and , the two-term exponential sum ( , , ; ) is defined as follows: where ( ) = 2 .
In this paper, we study the fourth power mean of the twoterm exponential sum ( , , ; ) as follows: where is any integer with ( , ) = 1.
Regarding this problem, it seems that none has studied it yet; at least we have not seen any related result before. The problem is interesting, because it can reflect that the mean value of ( , , ; ) is well behaved. The main purpose of this paper is to use the analytic methods and the properties of Gauss sums to study the special case of (4) with = 5, = , an odd prime and give an interesting identity and asymptotic formula for it. That is, we will prove the following conclusion. Theorem 1. Let > 3 be a prime. Then for any integer with ( , ) = 1, one has the identity where ( * / ) denotes the Legendre symbol. For any prime with 5 | − 1, one cannot give an exact computational formula in our theorem at present. The difficulty is that one needs to know the value of the character sums where 1 is any 5-order character mod . For any integer ℎ ≥ 3, whether there exists an exact computational formula for is an open problem, where is an odd prime and ( , ) = 1.

Several Lemmas
In this section, we will give several lemmas which are necessary in the proof of our theorem. In the proving process of all lemmas, we used many properties of Gauss sums; all these can be found in [1], so they will not be repeated here. First we have the following.

Lemma 2.
Letting be an odd prime with > 3, then one has the identity where ( * / ) denotes the Legendre's symbol.
Proof. For any prime , note that if passes through a complete residue system mod , then 2 + 1 also passes through a complete residue system mod , so note the identity (this formula can be found in Hua's book, Section 7.8, Theorem 8.2 [9]). One has This proves Lemma 2.

Lemma 4. Let be an odd prime and let be a 5th character mod . Then one has the identity
Proof. Noting that (−1) = 1, from the definition and properties of the classical Gauss sums, we have The Scientific World Journal Similarly, we also have This proves Lemma 4.

Proof of the Theorem
In this section, we shall complete the proof of our theorem. First from the orthogonality of characters mod we have On the other hand, if 5 † − 1, then any non-principal character is not a 5-order character mod . Note that From (22) and Lemma 3 we have If 5 † − 1, then combining (21) and (23) we may immediately deduce the identity The Scientific World Journal If 5 | − 1, since 1 ̸ = 0 is a 5-order character mod , 1 = 4 1 and 1 2 = 3 1 are also 5-order characters mod , then note that Now our theorem follows from (25) and (28).